manifolds and cobordisms
cobordism theory, Introduction
Given an embedding of manifolds $i : X \hookrightarrow Y$, the Thom collapse map is a useful approximation to its would-be left inverse. It is used to define pushforward of cohomology-classes along $i$ (“Umkehr maps”). It also appears as the key step in Thom's theorem.
All topological spaces in the following are taken to be compact.
Consider $X$ and $Y$ two manifolds and
an embedding.
Write
$N_i X \coloneqq i^* T Y/ T X$ for the normal bundle;
$Th(N_i X)$ for the Thom space of the normal bundle;
$f \colon N_i X \longrightarrow Y$ for any choice of tubular neighbourhood of $i$.
The collapse map (or the Pontrjagin-Thom construction) associated to $i$ and the choice of tubular neighbourhood $f$ is
where the first morphism is the projection onto the quotient and the second is the canonical homeomorphism to the Thom space of the normal bundle.
Since in the construction of remark every point of $N_i X$ is associated to a particular point of $X$, the collapse map lifts to a map
from $Y$ to the smash product of the Thom space (canonically regarded as a pointed topological space) and the topological space $X$ with a base point adjoined.
(e.g. Rudyak 98, p. 317)
Of particular interest is the case where $Y$ in the above is a Cartesian space $\mathbb{R}^{dim X + k}$ or rather its one-point compactification, the sphere $S^{dim X + k}$. By the Whitney embedding theorem, for every $n \in \mathbb{N}$ there exists an $k \in \mathbb{N}$ such that every manifold $X$ of dimension $n$ has an embedding $X \hookrightarrow \mathbb{R}^{n+k} \to S^{n+k}$. In this case the collapse map of def. has the form
Composing this further with the canonical map $N_i X \longrightarrow E O(k) \underset{O(k)}{\times} \mathbb{R}^{k}$ to the universal vector bundle of rank $k$ yields a map
from to the $k$th space in the Thom spectrum $M O$. This hence defines an element in the homotopy group $\pi_{k}(M O)$ of the Thom spectrum. Thom's theorem says that all elements in the homotopy groups of $M O$ arise this way, and that they retain precisely the information of the cobordism equivalence class of manifolds $X$.
In this case the refined Thom collapse map of def. is of the form
The refined map in example lifts to a morphism of spectra
where $\mathbb{S}$ denotes the sphere spectrum and $Th(N_i X)$ now the Thom spectrum of the normal bundle.
This morphism is the unit of an adjunction which exhibits the suspension spectrum $\Sigma_+^\infty X$ as a dualizable object in the stable homotopy category, with dual object $\Sigma^{-n-k} Th(N_i X)$. See at Atiyah duality and at n-duality.
Equivalently, one may proceed as follows. For a framed manifold i.e. a manifold $M^n$ with a chosen trivialization of the normal bundle $N_i (M^n)$ in some $\mathbf{R}^{n+r}$ one has $T N_i(M^n)\cong \Sigma^r(M^n_+)$ where $M^n_+$ is the union of $M^n$ with a disjoint base point. Identify a sphere $S^{n+r}$ with a one-point compactification $\mathbf{R}^{n+r}\cup \{\infty\}$. Then the Pontrjagin-Thom construction is the map $S^{n+r}\to Th(N_i X)$ obtained by collapsing the complement of the interior of the unit disc bundle $D(N_i M^n)$ to the point corresponding to $S(N_i M^n)$ and by mapping each point of $D(N_i M^n)$ to itself. Thus to a framed manifold $M^n$ one associates the composition
and its homotopy class defines an element in $\pi_{n+r}(S^r)$.
The following is a more abstract description of Pontryagin-Thom collapse in the stable homotopy theory of sphere spectrum-(∞,1)-module bundles.
Write
for the Spanier-Whitehead duality map which sends a topological space first to its suspension spectrum and then that to its dual object in the (∞,1)-category of spectra.
For $X$ a compact manifold, let $X \to \mathbb{R}^n$ be an embedding and write $S^n \to X^{\nu_n}$ for the classical Pontryagin-Thom collapse map for this situation, and write
for the corresponding looping map from the sphere spectrum to the Thom spectrum of the negative tangent bundle of $X$. Then Atiyah duality produces an equivalence
which identifies the Thom spectrum with the dual object of $\Sigma^\infty_+ X$ in $\mathbb{S} Mod$ and this constitutes a commuting diagram
identifying the classical Pontryagin-Thom collapse map with the abstract dual morphism construction of prop. .
More generally, for $W \hookrightarrow X$ an embedding of manifolds, then Atiyah duality identifies the Pontryagin-Thom collapse maps
with the abstract dual morphisms
Given now $E \in CRing_\infty$ an E-∞ ring, then the dual morphism $\mathbb{S} \to D X$ induces under smash product a similar Pontryagin-Thom collapse map, but now not in sphere spectrum-(∞,1)-modules but in $E$-(∞,1)-modules.
The image of this under the $E$-cohomology functor produces
If now one has a Thom isomorphism ($E$-orientation) $[D X \otimes_{\mathbb{S}} E, E] \simeq [X,E]$ that identifies the cohomology of the dual object with the original cohomology, then together with produces the Umkehr map
that pushes the $E$-cohomology of $X$ to the $E$-cohomology of the point. Analogously if instead of the terminal map $X \to \ast$ we start with a more general map $X \to Y$.
More generally a Thom isomorphism may not exists, but $[D X \otimes_{\mathbb{S}} E, E]$ may still be equivalent to a twisted cohomology-variant $[X,E]_{\chi}$ of $[X,E]$, namely to $[\Gamma_X(\chi),E]$, where $\chi \colon \Pi(X) \to E Line \hookrightarrow E Mod$ is an (flat) $E$-(∞,1)-module bundle on $X$ and and $\Gamma \simeq \underset{\to}{\lim}$ is the (∞,1)-colimit (the generalized Thom spectrum construction). In this case the above yields a twisted Umkehr map.
For given $i$ all collapse maps for different choices of tubular neighbourhood $f$ are homotopic.
By the fact that the space of tubular neighbourhoods (see there for details) is contractible.
For $X$ a closed smooth manifold of dimension $D$, the Pontryagin-Thom construction (e.g. Kosinski 93, IX.5) identifies the set
of cobordism classes of closed and normally framed submanifolds $\Sigma \overset{\iota}{\hookrightarrow} X$ of dimension $d$ inside $X$ with the cohomotopy $\pi^{D-d}(X)$ of $X$ in degree $D- d$
(e.g. Kosinski 93, IX Theorem (5.5))
In particular, by this bijection the canonical group structure on cobordism groups in sufficiently high codimension (essentially given by disjoint union of submanifolds) this way induces a group structure on the cohomotopy sets in sufficiently high degree.
The following terms all refer to essentially the same concept:
Lev Pontryagin, Smooth manifolds and their applications in homotopy theory. 1959 American Mathematical Society Translations, Ser. 2, Vol. 11 pp. 1–114 American Mathematical Society, Providence, R.I. (pdf, arXiv:10.1142/9789812772107_0001)
John Milnor, section 7 of Topology – From the differentiable viewpoint, 1965 (pdf)
Stanley Kochmann, section 1.5 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Yuli Rudyak, In Thom spectra, Orientability and Cobordism, Springer 1998 (pdf)
Cary Malkiewich, Unoriented cobordism and MO, 2011 (pdf)
An illustration is given on slide 15
Antoni Kosinski, chapter IX of Differential manifolds, Academic Press 1993 (pdf)
Ralph Cohen, John Klein, Umkehr Maps (arXiv:0711.0540)
Victor Snaith, Stable homotopy around the arf-Kervaire invariant, Birkhauser 2009
The general abstract formulation in stable homotopy theory is in sketched in section 9 of
and is in section 10 of
with an emphases on parameterized spectra.
Last revised on February 11, 2019 at 10:38:33. See the history of this page for a list of all contributions to it.